Finite Group Databasebeta

← #15926736

Subgroups — #15926736

Order 3328.

840 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117850C211yesmaximal, characteristic
2#13116403C211yesmaximal, characteristic, complemented
3#13116332C211yesmaximal, characteristic, complemented
4#13116221C211yesmaximal, characteristic, complemented
5#13115969C211yesmaximal, characteristic, complemented
6#13115785C211yesmaximal, characteristic, complemented
7#13115404C211yesmaximal, characteristic, complemented
8#13115272C211yesmaximal, characteristic
9#13115158C211yesmaximal, characteristic
10#13115082C211yesmaximal, characteristic, complemented
11#13115023C211yesmaximal, characteristic, complemented
12#13115021C211yesmaximal, characteristic, complemented
13#13114843C211yesmaximal, characteristic, complemented
14#13114557C211yesmaximal, characteristic, complemented
15C13×#2439C211yesmaximal, characteristic, complemented
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×#117C2211yescharacteristic
18C2×#88184C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6099C2211yescharacteristic
21C2×#88135—21no—
22C2×C4×D52C2211yescharacteristic
23C13×#396—21no—
24C13×#393—21no—
25C13×#391—21no—
26C13×#389C2211yescharacteristic
27C13×#389C2211yescharacteristic
28C13×#387—21no—
29C13×#382C2211yescharacteristic
30C13×#380C2211yescharacteristic
31C4×C13×C22⋊C4C2211yescharacteristic
32C2×C22⋊#1055C2211yescharacteristic
33C2×#88049C2211yescharacteristic
34C2×#88049C2211yescharacteristic
35C2×#88049C2211yescharacteristic
36C2×#88049C2211yescharacteristic
37C2×#88047C2211yescharacteristic
38C2×#88047C2211yescharacteristic
39#11718619—21no—
40#11718600C2211yescharacteristic
41C4×C4⋊Dic13C2211yescharacteristic
42#11718589C2211yescharacteristic
43C2×C4⋊#1055C2211yescharacteristic
44C2×C4⋊#1055C2211yescharacteristic
45#11718331—21no—
46#11718329—21no—
47#11718121C2211yescharacteristic
48#11718121C2211yescharacteristic
49#11718121C2211yescharacteristic
50#11718121C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C22×Q8×C13—21no—
60C23×C4×C13C2311yescharacteristic
61C22×C22⋊D13—21no—
62C23×Dic13—21no—
63C22×D52C2311yescharacteristic
64C22×C4×D13—21no—
65C13×C4⋊Q8—22no—
66C13×C4⋊Q8—41no—
67C13×#117—22no—
68C13×#117—41no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C22⋊C4—21no—
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4D411yescharacteristic
79C2×C13×C22⋊C4D411yescharacteristic
80C2×C42×C13—21no—
81C2×C42×C13C2311yescharacteristic
82C2×C42×C13C2311yescharacteristic
83#88184—22no—
84#88184Q814yescomplemented
85C2×#6103—21no—
86C2×#6103—21no—
87C2×#6103—21no—
88C2×#6103—21no—
89C2×C4⋊Dic13—22no—
90C2×C4⋊Dic13—21no—
91C2×C4⋊Dic13—41no—
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×#6101—21no—
96C2×C4×Dic13—21no—
97C2×#6099—22no—
98C2×#6099—41no—
99C2×#6099—41no—
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104#88135—44no—
105C4×D52—22no—
106C4×D52—41no—
107C13×#95—21no—
108C13×#95—21no—
109C13×#95—21no—
110C13×#95—21no—
111C13×#95—21no—
112C13×#95D411yescharacteristic
113C13×#95D411yescharacteristic
114#88071—21no—
115#88071—21no—
116C22⋊#1055—22no—
117#88049—22no—
118#88049—41no—
119#88049—41no—
120#88049—41no—
121#88047—22no—
122#88047—41no—
123C4⋊#1055—41no—
124C4⋊#1055—41no—
125#26426—131nomaximal, Sylow
126C24×C13C2×D411yescharacteristic
127C23×D13—21no—
128C2×Q8×C13—42no—
129C2×Q8×C13—42no—
130C2×Q8×C13—81no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2×Q812yes—
145C22×C4×C13#4112yes—
146C22×C4×C13C2411yescharacteristic
147C2×C22⋊D13—24no—
148C2×C22⋊D13—42no—
149C2×C22⋊D13—42no—
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—42no—
154C22×Dic13—42no—
155C22×Dic13—81no—
156C2×D52—21no—
157C2×D52—21no—
158C2×D52—21no—
159C2×D52—21no—
160C2×D52—21no—
161C2×D52C2×Q812yes—
162C2×C4×D13—42no—
163C2×C4×D13—42no—
164C2×C4×D13—81no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4—42no—
173C13×C4⋊C4—42no—
174C13×C4⋊C4—81no—
175C13×C4⋊C4#4112yes—
176C13×C22⋊C4—22no—
177C13×C22⋊C4—41no—
178C13×C22⋊C4—42no—
179C13×C22⋊C4—81no—
180C42×C13—21no—
181C42×C13—21no—
182C42×C13—21no—
183C42×C13—21no—
184C42×C13—42no—
185#6103—24no—
186#6103—42no—
187#6103—42no—
188#6103—81no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13—42no—
194C4⋊Dic13C2×Q812yes—
195C4⋊Dic13#4112yes—
196#6101—42no—
197C4×Dic13—42no—
198#6099—44no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211#2439D1311yescharacteristic, complemented
212#2361—131no—
213#2101—131no—
214#2087—131no—
215#2067—131no—
216#2023—131no—
217#1987—131no—
218#1918—131no—
219#1895—131no—
220#1875—131no—
221#1862—131no—
222#1852—131no—
223#1851—131no—
224#1804—131no—
225#1736—131no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13C22⋊Q812yes—
233C23×C13C22×D411yescharacteristic
234C22×D13—22no—
235C22×D13—41no—
236C22×D13—41no—
237C22×D13—41no—
238C22×D13—41no—
239C22×D13—41no—
240C22×D13—41no—
241Q8×C13—84no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—22no—
247C2×C4×C13—22no—
248C2×C4×C13—22no—
249C2×C4×C13—22no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—22no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—21no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13—41no—
269C2×C4×C13—41no—
270C2×C4×C13—41no—
271C2×C4×C13C22⋊Q812yes—
272C2×C4×C13AGL(1,ℤ/8)12yes—
273C2×C4×C13C22×Q811yescharacteristic
274C2×C4×C13C2×#4111yescharacteristic
275C2×C4×C13C2×#4111yescharacteristic
276C2×C4×C13C2×#4111yescharacteristic
277C2×C4×C13H(5,𝔽2)11yescharacteristic
278C2×C4×C13H(5,𝔽2)11yescharacteristic
279C22⋊D13—48no—
280C2×Dic13—22no—
281C2×Dic13—22no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—84no—
290D52—22no—
291D52—22no—
292D52—41no—
293D52—41no—
294C4×D13—84no—
295#1055—41no—
296#1055—41no—
297#1055—41no—
298#1055—41no—
299C22×M16—131no—
300C2×C4⋊Q8C2×D1311yescharacteristic
301C2×#121—131no—
302C2×C4×D4—131no—
303C2×C4×D4—261no—
304C2×#117C2×D1311yescharacteristic
305C22×C4⋊C4—131no—
306C2×#107—131no—
307C2×#107—131no—
308C2×C4⋊C8—131no—
309C2×C4⋊C8—131no—
310C2×#102—131no—
311C2×#102—131no—
312C2×#102—131no—
313C2×#102—131no—
314C2×C22⋊C8—131no—
315#396—21no—
316#393—21no—
317#391—21no—
318#390—131no—
319#389—131no—
320#389C2×D1311yescharacteristic
321#389C2×D1311yescharacteristic
322#388—261no—
323#387—21no—
324#386—261no—
325#385—261no—
326#382C2×D1311yescharacteristic
327#380C2×D1311yescharacteristic
328C4×C4⋊C4—131no—
329C4×C22⋊C4C2×D1311yescharacteristic
330#340—131no—
331#340—131no—
332#340—131no—
333#340—131no—
334#340—131no—
335#340—131no—
336#340—131no—
337#340—131no—
338#340—131no—
339#340—131no—
340#336—131no—
341#336—131no—
342C22×C13—22no—
343C22×C13—22no—
344C22×C13—22no—
345C22×C13—41no—
346C22×C13—41no—
347C22×C13—41no—
348C22×C13—41no—
349C22×C13C2×C22⋊Q811yescharacteristic
350C22×C13#54611yescharacteristic
351C22×C13#54611yescharacteristic
352C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
353C22×C13#57511yescharacteristic
354C22×C13#57611yescharacteristic
355C22×C13#57811yescharacteristic
356C2×D13—42no—
357C2×D13—42no—
358C2×D13—42no—
359C2×D13—81no—
360C2×D13—81no—
361C2×D13—81no—
362C2×D13—81no—
363C4×C13—22no—
364C4×C13—22no—
365C4×C13—41no—
366C4×C13—41no—
367C4×C13—42no—
368C4×C13—41no—
369C4×C13—81no—
370C4×C13—81no—
371C4×C13—81no—
372C4×C13#52911yescharacteristic
373C4×C13#53811yescharacteristic
374C4×C13#55411yescharacteristic
375C4×C13#55411yescharacteristic
376Dic13—42no—
377Dic13—42no—
378Dic13—81no—
379C22×Q8—21no—
380C22×D4—131no—
381C22×D4—261no—
382C23×C4—261no—
383C23×C4—261no—
384C23×C4C22×D1311yescharacteristic
385C2×M16—262no—
386C2×M16—521no—
387C2×M16—521no—
388C22×C8—131no—
389C22×C8—131no—
390C22×C8—131no—
391C22×C8—131no—
392C4⋊Q8—22no—
393C4⋊Q8—41no—
394#121—134no—
395#121—262no—
396C4×D4—262no—
397C4×D4—521no—
398C4×D4—524no—
399#117—22no—
400#117—41no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—131no—
405C2×C4⋊C4—131no—
406C2×C4⋊C4—131no—
407C2×C4⋊C4—262no—
408C2×C4⋊C4—261no—
409C2×C4⋊C4—261no—
410C2×C4⋊C4—521no—
411C2×C4⋊C4C22×D1311yescharacteristic
412C2×C4⋊C4C22×D1311yescharacteristic
413C2×C4⋊C4C22×D1311yescharacteristic
414C2×C4⋊C4C22×D1311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—261no—
418C2×C22⋊C4—261no—
419C2×C22⋊C4—261no—
420C2×C22⋊C4—261no—
421C2×C22⋊C4C22⋊D1311yescharacteristic
422C2×C22⋊C4C22⋊D1311yescharacteristic
423C2×C42—21no—
424C2×C42—261no—
425C2×C42C22×D1311yescharacteristic
426C2×C42C22×D1311yescharacteristic
427#107—262no—
428#107—521no—
429C4⋊C8—521no—
430C4⋊C8—521no—
431#102—262no—
432#102—521no—
433#102—521no—
434#102—521no—
435C22⋊C8—262no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—261no—
442#95—261no—
443#95C22⋊D1311yescharacteristic
444#95C22⋊D1311yescharacteristic
445C2×C13—42no—
446C2×C13#245011yescharacteristic
447C2×C13#307711yescharacteristic
448C2×C13#308311yescharacteristic
449C2×C13#330811yescharacteristic
450C2×C13#331311yescharacteristic
451C2×C13#332211yescharacteristic
452C2×C13#332511yescharacteristic
453D13—82no—
454C24—261no—
455C24C2×C22⋊D1311yescharacteristic
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—81no—
459C2×D4—132no—
460C2×D4—264no—
461C2×D4—261no—
462C2×D4—261no—
463C2×D4—261no—
464C2×D4—261no—
465C2×D4—261no—
466C2×D4—522no—
467C2×D4—522no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—261no—
477C22×C4—261no—
478C22×C4—261no—
479C22×C4—522no—
480C22×C4—522no—
481C22×C4—522no—
482C22×C4—522no—
483C22×C4—1041no—
484C22×C4—1041no—
485C22×C4Q8×D1312yes—
486C22×C4#613112yes—
487C22×C4C2×C22⋊D1311yescharacteristic
488C22×C4C2×C22⋊D1311yescharacteristic
489C22×C4C2×C22⋊D1311yescharacteristic
490C22×C4C2×C22⋊D1311yescharacteristic
491C22×C4C2×C22⋊D1311yescharacteristic
492C22×C4C23×D1311yescharacteristic
493M16—524no—
494C2×C8—261no—
495C2×C8—261no—
496C2×C8—261no—
497C2×C8—261no—
498C2×C8—261no—
499C2×C8—261no—
500C2×C8—261no—
501C2×C8—261no—
502C2×C8—261no—
503C2×C8—261no—
504C2×C8—261no—
505C2×C8—261no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—132no—
517C4⋊C4—132no—
518C4⋊C4—261no—
519C4⋊C4—261no—
520C4⋊C4—261no—
521C4⋊C4—261no—
522C4⋊C4—522no—
523C4⋊C4—522no—
524C4⋊C4#612912yes—
525C22⋊C4—22no—
526C22⋊C4—41no—
527C22⋊C4—42no—
528C22⋊C4—81no—
529C22⋊C4—264no—
530C22⋊C4—522no—
531C22⋊C4—522no—
532C22⋊C4—1041no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—42no—
538C42—522no—
539C13#2642611yescharacteristic, complemented, Sylow
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—262no—
547C23—521no—
548C23—521no—
549C23—521no—
550C23—521no—
551C23—521no—
552C23—521no—
553C23#8820012yes—
554C23C22×C22⋊D1311yescharacteristic
555Q8—84no—
556D4—262no—
557D4—262no—
558D4—528no—
559D4—521no—
560D4—521no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—262no—
591C2×C4—262no—
592C2×C4—521no—
593C2×C4—521no—
594C2×C4—521no—
595C2×C4—521no—
596C2×C4—521no—
597C2×C4—521no—
598C2×C4—521no—
599C2×C4—1044no—
600C2×C4—1044no—
601C2×C4#8820012yes—
602C2×C4#8820312yes—
603C2×C4C2×#612911yescharacteristic
604C2×C4#8825111yescharacteristic
605C2×C4C2×Q8×D1311yescharacteristic
606C2×C4C2×#613111yescharacteristic
607C2×C4#41×D1311yescharacteristic
608C2×C4#8825611yescharacteristic
609C8—521no—
610C8—521no—
611C8—521no—
612C8—521no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—522no—
621C22—522no—
622C22—522no—
623C22—1041no—
624C22—1041no—
625C22—1041no—
626C22—1041no—
627C22#1171951111yescharacteristic
628C22C2×#8820011yescharacteristic
629C22C2×#8820311yescharacteristic
630C22#1171952711yescharacteristic
631C22#1171953411yescharacteristic
632C22#1171954011yescharacteristic
633C22#1171954311yescharacteristic
634C4—22no—
635C4—22no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—522no—
644C4—522no—
645C4—1041no—
646C4#1171932911yescharacteristic
647C4#1171934011yescharacteristic
648C4#1171934111yescharacteristic
649C4#1171934911yescharacteristic
650C2—42no—
651C2—1042no—
652C2#1311865611yescharacteristic
653C2#1312462911yescharacteristic
654C2#1312468311yescharacteristic
655C2#1312682711yescharacteristic
656C2#1312689611yescharacteristic
657C2#1312703711yescharacteristic
658C2#1312707211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2439C2
socleC23×C13C22×D4
prime radicalC2×#88184C22
hypercenter#2439D13
hypocenterC13#26426
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #13116403C2semidirect
3. #13116332C2semidirect
4. #13116221C2semidirect
5. #13115969C2semidirect
6. #13115785C2semidirect
7. #13115404C2semidirect
10. #13115082C2semidirect
11. #13115023C2semidirect
12. #13115021C2semidirect
13. #13114843C2semidirect
14. #13114557C2semidirect
15. C13×#2439C2semidirect
84. #88184Q8semidirect
211. #2439D13semidirect
539. C13#26426semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#131178502
#131164032
#131163322
#131162212
#131159692
#131157852
#131154042
#131152722
#131151582
#131150822
#131150232
#131150212
#131148432
#131145572
C13×#24392
#2642613